From this coupling between the macro- and microscale equations, we get the classic multiscale formulation of the porous electrode model. In fact, the widely used Doyle-Fuller-Newman model is conceptually a homogenised model, even though in [1]}, [2]}, [3]} it was originally justified on an ad hoc basis, rather than being formally derived. However, asymptotic homogenisation offers us a tool to formally derive the homogenised model (including the equations and the effective parameters) from the full equations at the microscale in a systematic manner [4]}, [5]}, [6]}, [7]}, [8]}. Asymptotic homogenisation is a well-established mathematical technique and the technical details can be found in many books (e.g. [9]}, [10]}). The key idea of this method is that, if the model has two (or more) very distinct length scales (such as particles and whole electrodes in batteries), they can be treated as independent in terms of the spatial mathematical operators. This scale separation leads, through a systematic procedure, to the homogenised model. Homogenisation might appear analogous to the volume averaging method, [11]}, [12]} however, the latter requires an ad hoc closure condition, usually empirically defined, in order to complete the problem. As discussed in [13]}, in which a comparison is made between the two methodologies, the main differences are in the assumptions and formalisms of the two methods, rather than in the results they yield.
